Singular vector and singular subspace distribution for the matrix denoising model

Abstract

In this paper, we study the matrix denoising model $Y=S+X$, where $S$ is a low rank deterministic signal matrix and $X$ is a random noise matrix, and both are $M\times n$. In the scenario that $M$ and $n$ are comparably large and the signals are supercritical, we study the fluctuation of the outlier singular vectors of $Y$, under fully general assumptions on the structure of $S$ and the distribution of $X$. More specifically, we derive the limiting distribution of angles between the principal singular vectors of $Y$ and their deterministic counterparts, the singular vectors of $S$. Further, we also derive the distribution of the distance between the subspace spanned by the principal singular vectors of $Y$ and that spanned by the singular vectors of $S$. It turns out that the limiting distributions depend on the structure of the singular vectors of $S$ and the distribution of $X$, and thus they are nonuniversal. Statistical applications of our results to singular vector and singular subspace inferences are also discussed.